A torus is a surface shaped like a donut. Imagine that I've painted two rings on a torus. One is on the outer surface, and goes through the hole in the center, coming around from the other side. The other ring is on the inner surface, and goes all the way around the hole in the center. These two rings of paint are linked.

I then cut a small hole in the torus. Through this hole, I turn the torus inside-out.

In the process, the rings of paint switch from the outer surface to the inner surface and vice versa. Therefore, they have become unlinked. How?

This supposition is basically a proof that you cannot turn the torus inside out through a mere local small hole in the torus. To turn it inside out you must make a cut that would go through at least one member of any such pair of painted rings.

Posted by Charlie on 2007-01-02 08:40:00